Multiply the following complex numbers, marked as blue dots on the graph: $( e^{\pi i}) \cdot (5 e^{\pi i / 2})$ (Your current answer will be plotted in orange.)
Solution: Multiplying complex numbers in polar forms can be done by multiplying the lengths and adding the angles. The first number ( $ e^{\pi i}$ ) has angle $\pi$ and radius $1$ The second number ( $5 e^{\pi i / 2}$ ) has angle $\frac{1}{2}\pi$ and radius $5$ The radius of the result will be $1 \cdot 5$ , which is $5$ The angle of the result is $\pi + \frac{1}{2}\pi = \frac{3}{2}\pi$ The radius of the result is $5$ and the angle of the result is $\frac{3}{2}\pi$.